OFFSET
0,2
COMMENTS
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 23 ).
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 64 ).
Numbers k such that k and (k+1) have the same binary digital sum. - Benoit Cloitre, Jun 05 2002
Numbers k such that (1 + sqrt(k))/2 is an algebraic integer. - Alonso del Arte, Jun 04 2012
Numbers k such that 2 is the only prime p that satisfies the relationship p XOR k = p + k. - Brad Clardy, Jul 22 2012
1, 9, 21, 37, 57, 81, ...
5, 17, 33, 53, 77, 105, ...
13, 29, 49, 73, 101, 133, ...
25, 45, 69, 97, 129, 165, ...
41, 65, 93, 125, 161, 201, ...
61, 89, 121, 157, 197, 241, ...
...
- R. J. Mathar, Jul 10 2013
With leading term 2 instead of 1, 1/a(n) is the largest tolerance of form 1/k, where k is a positive integer, so that the nearest integer to (n - 1/k)^2 and to (n + 1/k)^2 is n^2. In other words, if interval arithmetic is used to square [n - 1/k, n + 1/k], every value in the resulting interval of length 4n/k rounds to n^2 if and only if k >= a(n). - Rick L. Shepherd, Jan 20 2014
Odd numbers for which the number of prime factors congruent to 3 (mod 4) is even. - Daniel Forgues, Sep 20 2014
For the Collatz conjecture, we identify two types of odd numbers. This sequence contains all the descenders: where (3*a(n) + 1) / 2 is even and requires additional divisions by 2. See A004767 for the ascenders. - Fred Daniel Kline, Nov 29 2014 [corrected by Jaroslav Krizek, Jul 29 2016]
a(n-1), n >= 1, is also the complex dimension of the manifold M(S), the set of all conjugacy classes of irreducible representations of the fundamental group pi_1(X,x_0) of rank 2, where S = {a_1, ..., a_{n}, a_{n+1} = oo}, a subset of P^1 = C U {oo}, X = X(S) = P^1 \ S, and x_0 a base point in X. See the Iwasaki et al. reference, Proposition 2.1.4. p. 150. - Wolfdieter Lang, Apr 22 2016
For n > 3, also the number of (not necessarily maximal) cliques in the n-sunlet graph. - Eric W. Weisstein, Nov 29 2017
For integers k with absolute value in A047202, also exponents of the powers of k having the same unit digit of k in base 10. - Stefano Spezia, Feb 23 2021
Starting with a(1) = 5, numbers ending with 01 in base 2. - John Keith, May 09 2022
REFERENCES
K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 150.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Colin Defant and Noah Kravitz, Loops and Regions in Hitomezashi Patterns, arXiv:2201.03461 [math.CO], 2022. Theorem 1.3.
L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 16.
Tanya Khovanova, Recursive Sequences
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
William A. Stein's The Modular Forms Database, PARI-readable dimension tables for Gamma_0(N)
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Hilbert Number
Eric Weisstein's World of Mathematics, Sunlet Graph
Wikipedia, Interval arithmetic
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
a(n) = A005408(2*n).
Sum_{n>=0} (-1)^n/a(n) = (1/(4*sqrt(2)))*(Pi+2*log(sqrt(2)+1)) = A181048 [Jolley]. - Benoit Cloitre, Apr 05 2002 [corrected by Amiram Eldar, Jul 30 2023]
G.f.: (1+3*x)/(1-x)^2. - Paul Barry, Feb 27 2003 [corrected for offset 0 by Wolfdieter Lang, Oct 03 2014]
(1 + 5*x + 9*x^2 + 13*x^3 + ...) = (1 + 2*x + 3*x^2 + ...) / (1 - 3*x + 9*x^2 - 27*x^3 + ...). - Gary W. Adamson, Jul 03 2003
a(n) = A004766(n-1). - R. J. Mathar, Oct 26 2008
a(n) = 2*a(n-1) - a(n-2); a(0)=1, a(1)=5. a(n) = 4 + a(n-1). - Philippe Deléham, Nov 03 2008
A056753(a(n)) = 3. - Reinhard Zumkeller, Aug 23 2009
a(n) = 8*n - 2 - a(n-1) for n > 0, a(0) = 1. - Vincenzo Librandi, Nov 20 2010
The identity (4*n+1)^2 - (4*n^2+2*n)*(2)^2 = 1 can be written as a(n)^2 - A002943(n)*2^2 = 1. - Vincenzo Librandi, Mar 11 2009 - Nov 25 2012
A089911(6*a(n)) = 8. - Reinhard Zumkeller, Jul 05 2013
a(n) = A004767(n) - 2. - Jean-Bernard François, Sep 27 2013
From Ilya Gutkovskiy, Jul 29 2016: (Start)
E.g.f.: (1 + 4*x)*exp(x).
a(n) = Sum_{k = 0..n} A123932(k).
a(A005098(k)) = x^2 + y^2.
Inverse binomial transform of A014480. (End)
Dirichlet g.f.: 4*Zeta(-1 + s) + Zeta(s). - Stefano Spezia, Nov 02 2018
EXAMPLE
From Leo Tavares, Jul 02 2021: (Start)
Illustration of initial terms:
o
o o
o o o
o o o o o o o o o o o o o o o o
o o o
o o
o
(End)
MAPLE
seq(4*k+1, k=0..100); # Wesley Ivan Hurt, Sep 28 2013
MATHEMATICA
Range[1, 237, 4] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
Table[4 n + 1, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
4 Range[0, 20] + 1 (* Eric W. Weisstein, Nov 29 2017 *)
LinearRecurrence[{2, -1}, {5, 9}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[(1 + 3 x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)
PROG
(Magma) [n: n in [1..250 by 4]];
(Haskell)
a016813 = (+ 1) . (* 4)
a016813_list = [1, 5 ..] -- Reinhard Zumkeller, Feb 14 2012
(PARI) a(n)=4*n+1 \\ Charles R Greathouse IV, Mar 22 2013
(PARI) x='x+O('x^100); Vec((1+3*x)/(1-x)^2) \\ Altug Alkan, Oct 22 2015
(Scala) (0 to 59).map(4 * _ + 1) // Alonso del Arte, Aug 08 2018
(GAP) List([0..70], n->4*n+1); # Muniru A Asiru, Aug 08 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved